J Eng Math DOI 10.1007/s10665-016-9853-y A study on the onset of thermally modulated Darcy–Bénard convection Om P. Suthar · P. G. Siddheshwar · B. S. Bhadauria Received: 31 March 2015 / Accepted: 4 March 2016 © Springer Science+Business Media Dordrecht 2016 Abstract A stability analysis of linearized Rayleigh–Bénard convection in a densely packed porous layer was performed using a matrix differential operator theory. The boundary temperatures were assumed to vary periodically with time in a sinusoidal manner. The correction in the critical Darcy–Rayleigh number was computed and depicted graphically. It was shown that the phase difference between the boundary temperatures rather than the frequency of modulated temperatures decides the nature of influence of modulation on the onset of convection. Conclusions were drawn regarding the possible transitions from harmonic to subharmonic solutions. The results on the onset of thermally modulated convection in a rectangular porous enclosure were obtained using those on the modulated Darcy–Bénard convection. Keywords Darcy–Bénard convection · Matrix differential operator · Rectangular enclosure · Subharmonic instability · Temperature modulation Mathematics Subject Classification 76E06 · 76E15 · 76M45 · 76R10 · 76S05 1 Introduction Natural convection can occur in a fluid-saturated porous medium if the buoyancy forces within the fluid are suffi- ciently strong to overcome viscous drag. The classical Horton–Rogers–Lapwood problem [1, 2] considers an infinite horizontal porous layer with the lower surface maintained at a higher temperature than that of the upper surface (Fig. 1a). They found that convection occurred if the dimensionless Darcy–Rayleigh number (defined later in the paper) exceeds the critical value of 4π 2 . Following on from this, results are found for the time-independent boundary conditions of a Dirichlet or Neumann or Robin type [3, 4]. An additional small-amplitude temperature modulation O. P. Suthar (B ) Department of Mathematics, School of Advanced Sciences, VIT University, Vellore 632014, India e-mail: ompsuthar@gmail.com; ompsuthar@vit.ac.in P. G. Siddheshwar Department of Mathematics, Bangalore University, Jnanabharathi Campus, Bangalore 560 056, India B. S. Bhadauria Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India 123